\[{{}_{2}F_{1}}\left({a,b\atop c};\begin{bmatrix}t_{1}&0\\
0&t_{2}\end{bmatrix}\right)=\sum_{k=0}^{\infty}\frac{{\left(a\right)_{k}}{%
\left(c-a\right)_{k}}{\left(b\right)_{k}}{\left(c-b\right)_{k}}}{k!\,{\left(c%
\right)_{2k}}{\left(c-\tfrac{1}{2}\right)_{k}}}\*(t_{1}t_{2})^{k}{{}_{2}F_{1}}%
\left({a+k,b+k\atop c+2k};t_{1}+t_{2}-t_{1}t_{2}\right).\]